Maintainer's Corner

Readme for markov-realization-0.3.2

Markov Tutorial

Let Xₙ denote the nth state of a Markov chain with state space ℕ.
For x ≠ 0 define transition probabilities

p(x,0) = q,

p(x,x) = r, and

p(x,x+1) = s.

When x = 0, let
p(x,0) = q+r,
p(x,x+1) = s.
Let p(x,y) = 0 in all other cases.
Suppose we wanted to find
P[Xₙ = j ∩ d = k],
where d denotes the number of transitions from a positive integer to zero.
There are three values we need to track —
extinctions, probability, and state.
Extinctions add a value to a counter each time they happen
and the counter takes integral values,
so they can be represented by Sum Int.
Probabilities are multiplied each step,
and added when duplicate steps are combined.
We want decimal probabilities, so
we can represent this with Product Rational.
We will make a new type for the state.